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Question
If p, q and r in continued proportion, then prove the following :
`"p"^2 - "q"^2 + "r"^2 = "q"^4 (1/"p"^2 - 1/"q"^2 - 1/"r"^2)`
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Solution
p : q :: q : r ⇒ q2 = pr
`"p"^2 - "q"^2 + "r"^2 = "q"^4 (1/"p"^2 - 1/"q"^2 - 1/"r"^2)`
RHS
`"q"^4 (1/"p"^2 - 1/"q"^2 - 1/"r"^2)`
`= "q"^4 (("q"^2"r"^2 - "p"^2 "r"^2 + "p"^2"q"^2)/("p"^2"q"^2"r"^2))`
`= "q"^4 "q"^2 (("r"^2 - "q"^2 + "p"^2)/("q"^2"q"^4))`
= p2 - q2 + r2 = LHS
LHS = RHS. Hence, proved.
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